HARMONIC MEASURE, TRUE TREES and QUASICONFORMAL FOLDING Christopher Bishop, Stony Brook International Congress of Mathematicians
HARMONIC MEASURE, TRUE TREES AND QUASICONFORMAL FOLDING Christopher Bishop, Stony Brook International Congress of Mathematicians August 6, 2018 Rio de Janeiro, Brazil www.math.sunysb.edu/~bishop/lectures Harmonic measure = hitting distribution of Brownian motion Suppose Ω is a planar Jordan domain. Harmonic measure = hitting distribution of Brownian motion Let E be a subset of the boundary, ∂Ω. Harmonic measure = hitting distribution of Brownian motion Choose an interior point z ∈ Ω. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) ≈ 64/100. What if we move starting point z? Harmonic measure = hitting distribution of Brownian motion Different z gives ω(z,E, Ω) ≈ 12/100.
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